Skip to content

Commit

Permalink
galois_color
Browse files Browse the repository at this point in the history
  • Loading branch information
valbert4 committed Sep 12, 2024
1 parent 2babd8a commit c6e6708
Show file tree
Hide file tree
Showing 5 changed files with 48 additions and 12 deletions.
Original file line number Diff line number Diff line change
Expand Up @@ -75,7 +75,7 @@ relations:
See Ref. \cite{arxiv:2112.13617} for an alternative non-CSS extension of 2D color codes.'
- code_id: qudit_color
detail: 'Modular-qudit 2D color codes reduce to 2D color codes for \(q=2\).'
- code_id: galois_topological
- code_id: galois_color
detail: 'Galois-qudit 2D color codes reduce to 2D color codes for \(q=2\).'
- code_id: quantum_double_abelian
detail: |
Expand Down
Original file line number Diff line number Diff line change
Expand Up @@ -32,9 +32,7 @@ features:
relations:
parents:
- code_id: tqd_abelian
detail: 'The anyon theory corresponding to Abelian quantum double codes is defined by an Abelian group and trivial cocycle.
All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:code-switching]{condensing} certain bosonic anyons \cite{arxiv:2211.03798}.
Upon gauging some symmetries \cite{arxiv:1202.3120,arxiv:1605.01640,arxiv:1806.08679,arxiv:1806.08679}, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models \cite{arxiv:hep-th/9511195,arxiv:1508.03468,arxiv:2112.12757}.'
detail: 'The anyon theory corresponding to Abelian quantum double codes is defined by an Abelian group and trivial cocycle. All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and \hyperref[topic:code-switching]{condensing} certain bosonic anyons \cite{arxiv:2211.03798}. Upon gauging some symmetries \cite{arxiv:1202.3120,arxiv:1605.01640,arxiv:1806.08679,arxiv:1806.08679}, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models \cite{arxiv:hep-th/9511195,arxiv:1508.03468,arxiv:2112.12757}.'
- code_id: quantum_double
detail: 'The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.'

Expand Down
Original file line number Diff line number Diff line change
Expand Up @@ -12,7 +12,7 @@ introduced: '\cite{arxiv:1503.08800}'
# assumed to be lattice code, diff from code_id:color, as motivated by paper

description: |
An extension of color codes on lattices to modular qudits.
Extension of the color code to lattices of modular qudits.
Codes are defined analogous to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizer commute.
This can be done by puncturing a hyperspherical lattice \cite{arxiv:1311.0879} or constructing a star-bipartition; see \cite[Sec. III]{arxiv:1503.08800}.
Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and/or any twist defects present.
Expand Down
Original file line number Diff line number Diff line change
Expand Up @@ -3,27 +3,29 @@
## https://github.com/errorcorrectionzoo ##
#######################################################

code_id: galois_topological
code_id: galois_color
physical: galois
logical: galois

name: 'Galois-qudit topological code'
introduced: '\cite{arxiv:quant-ph/0609070,doi:10.1109/CIG.2010.5592860,arxiv:1202.3338}'
name: 'Galois-qudit color code'
introduced: '\cite{doi:10.1109/CIG.2010.5592860}'

alternative_names:
- '\(\mathbb{F}_q\)-qudit topological code'
- '\(\mathbb{F}_q\)-qudit color code'

description: |
Abelian topological code, such as a 2D surface \cite{arxiv:quant-ph/0609070,arxiv:1202.3338} or 2D color \cite{doi:10.1109/CIG.2010.5592860} code, constructed on lattices of Galois qudits.
Extension of the color code to 2D lattices of Galois qudits.
relations:
parents:
- code_id: galois_css
- code_id: 2d_stabilizer
- code_id: topological_abelian
- code_id: generalized_homological_product_css
- code_id: quantum_double
detail: 'A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.'
cousins:
- code_id: quantum_double_abelian
detail: 'A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Galois-qudit topological surface and color codes yield Abelian quantum-double codes via this decomposition.'
detail: 'A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.'


# Begin Entry Meta Information
Expand Down
Original file line number Diff line number Diff line change
@@ -0,0 +1,36 @@
#######################################################
## This is a code entry in the error correction zoo. ##
## https://github.com/errorcorrectionzoo ##
#######################################################

code_id: galois_topological
physical: galois
logical: galois

name: 'Galois-qudit surface code'
introduced: '\cite{arxiv:quant-ph/0609070,arxiv:1202.3338}'

alternative_names:
- '\(\mathbb{F}_q\)-qudit surface code'

description: |
Extension of the surface code to 2D lattices of Galois qudits.
relations:
parents:
- code_id: galois_css
- code_id: 2d_stabilizer
- code_id: generalized_homological_product_css
- code_id: quantum_double
detail: 'A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Galois-qudit surface codes yield Abelian quantum-double codes with \(GF(p^m)\cong \mathbb{Z}_p^m\) topological order via this decomposition.'
cousins:
- code_id: quantum_double_abelian
detail: 'A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits \cite{doi:10.1109/18.959288}\cite[Sec. 5.3]{arxiv:quant-ph/0501074}\cite{preset:GottesmanBook}. Galois-qudit surface codes yield Abelian quantum-double codes with \(GF(p^m)\cong \mathbb{Z}_p^m\) topological order via this decomposition.'


# Begin Entry Meta Information
_meta:
# Change log - most recent first
changelog:
- user_id: VictorVAlbert
date: '2022-07-27'

0 comments on commit c6e6708

Please sign in to comment.